Jean-Pierre Serre

1985 Balzan Prize for Mathematics

For his pioneering work in and major contributions to algebraic topology, algebraic geometry and number theory, in particular, for his original in-depth renewal of the foundations and techniques of algebraic topology and geometry.

The first work of Jean-Pierre Serre, which at once established his high reputation, concerned the application of homology to homotopy. In order to compute homotopy groups of certain usual spaces (such as the spheres), he had the daring idea to use auxiliary infinite dimensional spaces (loop spaces, Eilenherg-Mac Lane complexes) and to apply to them powerful homological methods which had just been developed. That required a degree of skill and astuteness which, after many years, remains astonishing. He also introduced the «localisation» process consisting in working «modulo» certain classes of abelian groups, thus laying the foundations of what would become the theory of rational homotopy types.

After having contributed to the cohomological interpretation of results of Henri Cartan un coherent analytic sheaves, having proved, together with Cartan, the fundamental finiteness theorem in the compact case, and having transposed the theory of Stein spaces to the projective situation, he applied these same methods to «abstract» algebraic geometry. In that way, he completely renovated the foundations of that science and opened the way to the monumental work of A. Grothendieck. His remarkable boldness was here, at the cost of considerable technical difficulties, to apply the methods of algebraic topology to topological spaces which were no Hausdorff (algebraic varieties endowed with the Zariski topology). Among the essential tools constantly used in the homological study of algebraic varieties, many are results of Serre.

His contributions, more recent, to number theory are also extremely diverse and innovating. Here are just a few of the numerous domains which he has enriched in the course of his incessant activity: «geometric» class field theory, Galois cohomology his published course remains the basic reference on the subject), arithmetic groups and their cohomology (occasion of a fruitful excursion in the theory of discrete groups and their action un trees), points of finite order of elliptic curves and 1-adic representations, modular forms, «effective» asymptotic formulas in analytic number theory, number of points of curves over finite fields. Besides their intrinsic importance, his contributions to each one of those topics have originated considerable research activity,

Jean-Pierre Serre’s mathematical thought is far from being confined to his published work. Many mathematicians in the whole world have been deeply influenced by him, through personal contacts, and the — often far-reaching — developments initiated by «questions of Sette» are innumerable. Furthermore, his talent as an expositor and the elegance of his style, universally considered as a model, are so exceptional that they, alone, can be viewed as a major contribution to contemporary mathematics.

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